If \(46 = 16 + 2(\mathrm{x} - 8)\), what is the value of \(2(\mathrm{x} - 8)\)?

1. TRANSLATE the problem requirements

• Given equation: \(\mathrm{46 = 16 + 2(x - 8)}\)
• Question asks for: the value of \(\mathrm{2(x - 8)}\)
• Key insight: We don't need to find x first—we can find \(\mathrm{2(x - 8)}\) directly

2. INFER the most efficient approach

• Since \(\mathrm{2(x - 8)}\) appears as a complete unit in the equation, isolate it directly
• This avoids unnecessary algebraic steps and reduces error potential

3. SIMPLIFY by isolating the target expression

• Subtract 16 from both sides of the equation:
  \(\mathrm{46 - 16 = 16 + 2(x - 8) - 16}\)
• Calculate: \(\mathrm{30 = 2(x - 8)}\)

Answer: C. 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students focus on "solving the equation" and assume they must find x first, missing that the question asks for \(\mathrm{2(x - 8)}\) directly.

They expand \(\mathrm{2(x - 8) = 2x - 16}\), substitute into the equation to get \(\mathrm{46 = 16 + 2x - 16}\), then solve \(\mathrm{46 = 2x}\) to get \(\mathrm{x = 23}\). But then they incorrectly think the answer is 23 instead of calculating \(\mathrm{2(23 - 8) = 2(15) = 30}\).

This may lead them to select Choice B (23).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need to subtract 16 from both sides but make an arithmetic error calculating \(\mathrm{46 - 16}\).

If they miscalculate and get 38 instead of 30, this leads them to select Choice D (38).

The Bottom Line:

The key challenge is recognizing that algebraic expressions can often be evaluated as complete units without breaking them down into individual variables—a strategic insight that makes complex problems surprisingly simple.